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Assume I’m given a standing wave function and I’m asked to find the phase velocity of the two waves that interfered and created the standing wave. Now, I have two questions:

  1. Is it possible to move faster than the phase velocity?

  2. Is it possible that the phase velocity is bigger than light velocity?

I just started to learn about wave mechanics so I'll be glad for a profound explanation for those questions.

I’m pretty sure this is a theoretical general question and it’s not related to the actual wave, but just in case I'll write the wave function:

$y(x,t)=0.04\sin(5\pi x+\alpha)\cos(40\pi t)$

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Take a plane wave:

$$ A(x, t) = Ae^{i\phi(x, t)} = Ae^{i(kx-\omega t)}$$

where

$$ \phi(x, t) = kx-\omega t$$

is the phase. The time derivative of the phase,

$$ \frac{\partial \phi}{\partial t} = -\omega $$

gives the frequency. Meanwhile, the spatial derivative

$$ \frac{\partial \phi}{\partial x} = k $$

is the wavenumber:

$$ k = \frac{2\pi}{\lambda} $$

The phase velocity is their ratio:

$$ v_{ph} = \frac{\omega} k $$

So that answer to (1) is yes.

In the above wave:

$$ \omega(k) = v_{ph} k$$

This is called the dispersion relation, or: how does the frequency depend on wavenumber?

The linear relation is called "dispersionless": all frequencies propagate at the same speed.

For a non-linear relation, such as:

$$ \omega(k) = \sqrt{(ck)^2 + (mc^2)^2} $$

then:

$$ v_{ph} = \frac{\sqrt{(ck)^2 + (mc^2)^2}} k = c\sqrt{1+\frac{m^2c^4}{k^2}} > c$$

which is larger than $c$. So the answer to (2) is "yes".

You should convince yourself that the phase $\phi(x, t)$ is local, that is, as it changes and say, the peak, or the zero crossing, moves: no information is being transferred (really: nothing more than an apparent position is "moving").

Energy (or information) travels at the group velocity:

$$ v_{gp} = \frac{d\omega}{dk} $$

which in the example given, is:

$$ v_{gp} = c\frac k {\omega} < c$$

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  • $\begingroup$ I didnt get how this answers (1) ? I knew that $ \omega=v_{ph}k $ but how does it shows that it is possible to move faster than this velocity? $\endgroup$ – FreeZe Oct 23 '20 at 21:38
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  1. Yes, the group velocity of a non-relativistic matter wave is twice the phase velocity.

  2. Yes, the phase velocity of a relativistic matter wave is always faster than light.

For more details, see e.g. my Phys.SE answer here.

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